My favorite book, that I strongly recommend despite the high price of around $100 in the US is "Mathematical Proofs" by Chartrand. You can get an international copy off eBay for around $45.
If you're weak on basic algebra etc, then you should instead start with "engineering mathematics" by Stroud, which has a foundations section that I started with several years ago when I started relearning math. It's designed for self-study.
I actually did find it helpful to do classes, I found most of the lower division math classes available online (i.e. calculus 1,2,3 and linear algebra). Sometimes, it helps to have deadlines, exams etc :)
Btw, if anyone out there already has a non math degree, but wants to study upper division and graduate level math formally, it turns out the way that is usually done on the US is to apply to a Math Masters program for "conditional admission" to the masters programs. They admit you, and then you do the upper division undergrad courses first, then move onto the masters programs. It's also possible to sign up for one-off classes at various universities via some kind of "open university" program, which is much easier to get into than formal admission to a degree course- I'm actually starting an Analysis course and a Linear algebra course at Berkeley tomorrow, as part of their "summer session", and you basically just sign up, pay your money, and turn up :)
Feel free to get in touch if anyone has any questions (email in profile)
This feeling that you are not armed for the subject is because there is a lot of dependent information between what you know and subjects like category theory and abstract algebra. Since you just got outta high school, you still have a lot to learn between where you are and where you want to be. Do not let that dissuade you tho, you can learn it, just gotta start.
Both MIT[1] and Stanford[2] have category theory as a graduate level course. I was not a math major but I assume that means you're like 4+ years away from learning this on the college track. Now, do not take that as a personal endorsement for going to college, you do you.
But, you are on hacker news, so I assume you want to learn, Well here is the MIT undergrad pure math major class requirements[3]. Its a good place to start learning an undergrad amount of math, the internet has resources everywhere to learn this stuff, it just takes time. Lots and lots of time.
One more tip, there is a trade-off between how hard something is to learn and how quickly you can learn it [4]. Do not over exert yourself too far in the difficult to learn direction, because you will become frustrated. Try and find a spot that is still fun, but not too fun, because then you are not maximizing your learning potential, assuming that is your goal. Learning how to learn can be very helpful, maximize your gains.
Also shout out to Numberphile on Youtube [5]. If you like math, you will like the channel.
[1] https://ocw.mit.edu/courses/mathematics/18-s996-category-the... [2] http://math.stanford.edu/~vakil/10-210A/ [3] https://math.mit.edu/academics/undergrad/major/course18/pure... [4] http://fancyfishgames.com/img/difficulty_curve.png [5] https://www.youtube.com/user/numberphile
http://abstract.ups.edu/download/aata-20160809.pdf try that for size.
Another good resource, except for the latter parts only being obviously useful for physics: http://www.staff.science.uu.nl/~gadda001/goodtheorist/primar...
Personally, I only started to enjoy math when I started hanging out with PhD students (in engineering as I was an engineer). They showed me what you can do with upper level math and that motivated me to learn it. I discovered that most math isn't like high school at all and is way cooler than I imagined.